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arXiv:1405.0420v1 [quant-ph] 2 May 2014
Journal of Nonlinear Optical Physics & Materials
c World Scientific Publishing Company
OPTIMUM TOPOLOGY OF QUASI-ONE DIMENSIONAL
NONLINEAR OPTICAL QUANTUM SYSTEMS.
RICK LYTEL, SHORESH SHAFEI∗ , and MARK G. KUZYK
Department of Physics and Astronomy, Washington State University
Pullman, Washington 99164-2814
[email protected]
∗ Present
Address: Department of Chemistry, Duke University,
Durham, North Carolina 27708-0346
We determine the optimum topology of quasi-one dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs
endowed with a metric and a multiparticle Hamiltonian acting on the edges, and have a
long application history in aromatic compounds, mesoscopic and artificial materials, and
quantum chaos. Quantum graphs have recently emerged as models of quasi-one dimensional electron motion for simulating quantum-confined nonlinear optical systems. This
paper derives the nonlinear optical properties of quantum graphs containing the basic
star vertex and compares their responses across topological and geometrical classes. We
show that such graphs have exactly the right topological properties to generate energy
spectra required to achieve large, intrinsic optical nonlinearities. The graphs have the
exquisite geometrical sensitivity required to tune wave function overlap in a way that optimizes the transition moments. We show that this class of graphs consistently produces
intrinsic optical nonlinearities near the fundamental limits. We discuss the application
of the models to the prediction and development of new nonlinear optical structures.
1. Introduction
The field of nonlinear optics has spawned decades of fundamental and applied
research on the interactions of strong electromagnetic fields with naturally occurring solid,1,2,3,4 liquid,5,6,7,8 liquid crystal,9,8 or gaseous materials,10,11 as
well as photonic crystals, mesoscopic solid state wires,12,13,14 and other artificial
systems.15,16,17,18 Research and commercial developments have driven the search
for new materials from which large effects may be extracted with ever-decreasing
field strengths and optical intensities. Ultrafast response is desirable in so many
applications that an entire field of research centers on off-resonant effects, whereby
photons interact with the material by exciting virtual transitions. This paper focuses exclusively on a new class of quantum structures for nonlinear optics called
quantum graphs, having those energy spectra and wavefunctions required to maximize the off-resonance nonlinear optical response, approaching the fundamental
limits allowed by quantum theory.19
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1.1. Ultrafast nonlinear optics
Nonlinear optical materials are quantum systems with polarizabilities that are nonlinear functions of external electromagnetic fields. Harmonic generation 20,21,22 ,
electro-optics 23 , saturable absorption 24 , phase conjugation 25,26 , four-wave mixing 27,28 , optical bistability 29,30 , ultrafast optics 31 , and waveguide switching 32,33
are among the many processes in NLO materials 34,35,36 of interest in communications, instrumentation, networking, image processing, and many other fields 37,38 .
The polarization vector for a general system is a complex function of every
allowed transition moment for the material, including electronic, vibrational, and
rotational transitions, and their corresponding transition energies and damping
factors. It is often expressed as a power series in the contractions of the nth order
susceptibility tensor with n-1 electric field components
Pi = αij Ej + βijk Ej Ek + γijkl Ej Ek El + ...
(1.1)
where αij is the linear polarizability tensor, βijk is the first hyperpolarizability
tensor, γijkl is the second hyperpolarizability tensor, and the Ej are the external
electromagnetic field components to which the material structure couples. For bulk
materials, the polarization expansion provides a means to measure the symmetry
properties of the susceptibilities and their bulk values. On the molecular level, the
expansion describes the response of a single structure to external optical fields.
Ultrafast nonlinear optical effects are created when the external fields are offresonance with the energy levels of the system. The hyperpolarizability tensors
become fully symmetric and their magnitudes and rotation properties are entirely
determined by the quantum system energy level differences Enm ≡ En − Em and
transition moments rnm ≡ hn|r|mi, where En are the energies and n, m are state
numbers. They may be calculated using a sum over states. For the x-diagonal tensor
elements, we have39
βxxx = −3e3
X′ x0n x̄nm xm0
n,m
(1.2)
En0 Em0
and

γxxxx = 4e4 
X ′ x0n x̄nm x̄ml xl0
n,m,l
En0 Em0 El0
−
X′ |x0n |2 |x0m |2
n,m
2 E
En0
m0


(1.3)
where x̄nm ≡ xnm − δnm x00 , and the prime on the sums indicates that the ground
state n = 0 is not included. These will be generalized later to multidimensional
space.
The experimentalist desires materials with maximum response in order to minimize the required optical field strength. Materials are comprised of nonlinear optical
structures, or moieties having microscopic hyperpolarizabilities that are calculated
from Eq. (1.2) and Eq. (1.3). The size of the moiety may be used to increase its
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response, but this then limits the number of such moieties that may be incorporated into a material. It is therefore desirable to create a size-independent metric
describing the intrinsic nonlinearities.
1.2. Intrinsic response
Scale-free, intrinsic hyperpolarizability tensors in the off-resonance regime may be
created by normalizing them to their maximum values. That maxima exist for a
given moiety is not immediately clear from the sum over states expressions, but the
quantum mechanics of the system provides an additional set of relations among the
spectra and transition moments, the Thomas-Reiche-Kuhn sum rules 40 , viz.,
Snm =
∞
X
p=0
[2Ep0 − (En0 + Em0 )] xnp xpm =
~2 N
δnm .
m
(1.4)
These constraints set fundamental limits on the hyperpolarizabilities 41,42,43,44 . The
fundamental limits depend only on the number of electrons, N , and the energy gap
between the ground and the first excited state, E10 . They were first derived by
truncating the sum over states Eq. (1.2) to two levels to calculate a maximum β,
and showing that the addition of a state to the two-level model always reduces the
value of β. Similar remarks hold for γ but with a three level model. The maxima
for the hyperpolarizabilities are then
3 3/2
N
e~
(1.5)
βmax = 31/4
7/2
m1/2
E10
and
γmax = 4
e 4 ~4
m2
N2
5 .
E10
(1.6)
Scale-independent (intrinsic) tensors are created by normalizing each tensor to
the fundamental limit. The fundamental limits are the highest attainable first and
second hyperpolarizabilities. Throughout this paper, all tensor components of the
hyperpolarizabilities are normalized by these maxima, ie,
γijkl →
γijkl
γmax
βijkl →
βijkl
.
βmax
(1.7)
The second hyperpolarizability normalized this way has a largest negative value
equal to −(1/4) of the maximum value. Hyperpolarizabilities normalized this way
enable direct comparisons of the intrinsic response without regard to size.
1.3. Toward an optimum energy spectrum
It is apparent that structures having the largest nonlinearities will have some set
of physical properties making their spectra and transition moments optimum for
maximizing the hyperpolarizabilities through the sum over states expressions Eq.
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(1.2) and Eq. (1.3). Quantum systems have yet to be found that achieve the maximum allowed values of the response. In fact, a factor of thirty gap existed between
theoretical and experimental values of the first hyperpolarizability prior to 2006 19 .
More recent developments have led to molecules with record hyperpolarizabilities
45,46,47,48
, still well short of the fundamental limits. Apparently the spectra and
transition moments (En0 , xnm ) for all of these moieties are far from optimum.
This observation is borne out through theoretical explorations of the behavior
required of states and spectra for a general quantum system using Monte Carlo
methods, constrained by the full TRK sum rules 49,40,50 . Monte Carlo studies of
large numbers of constrained sets of spectra and moments reveal that the optimum
spectrum for a system scales quadratically or faster with state number (which we
call a superscaling spectrum) 51 . Fig. 1 shows how the intrinsic hyperpolarizabilities
scale with the state scaling exponent k, defined through the scaling of the energy
En with state number n as En ∼ nk . Topological properties of a system determine
the spectra, while geometrical properties determine projections of the transition
moments onto a specific external axis. An optimum topology is one which produces
a superscaling spectrum.
-3
-2
-1
0
1
2
1.0
-3
1.0
-2
-1
0
1
2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.8
xxxx
xxx
max
0.8
min
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
0.0
-0.2
0.0
-3
-2
-1
0
1
Energy scaling exponent k
2
-3
-2
-1
0
1
2
Energy scaling exponent k
Fig. 1. Scaling of the maximum intrinsic first hyperpolarizability (left) and the maximum and
minimum intrinsic second hyperpolarizability with the scaling exponent k, where the energy spectrum scales with state number n as En ∼ nk .
The Monte Carlo results are precise, but a heuristic from the theory of fundamental limits will be shown to corroborate them. The three-level Ansatz (TLA),
which posits that only three states contribute for a system with a nonlinearity close
to its limit, is consistent with all observations and analysis to date, and is a suitable
heuristic to consider:
3/2
2
7/2 |x01 | x̄11
3L
3/4 m
(1.8)
E
βxxx → βxxx = 3
10
2
~2
E10
|x02 |2 x̄22
x01 x20 x12
+
+
+ c.c.
2
E20
E10 E20
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(with an equivalent expression for γxxxx , but requiring four states). Note that the
TLA does not imply that only three states are required to obey the sum rules.
Contributions of low lying states may dominate the hyperpolarizabilities, but contributions from large numbers of states to the sum rules are usually required. All
known models with a given topology employing a potential energy satisfy the TLA
when their geometries are tuned to yield the maximum hyperpolarizabilities for
their topology.52
Suppose we now simplify the three level model by replacing the products of
moments in Eq. (1.8) with expressions computed from the lowest four sum rule
constraints Eq, (1.4) truncated to only three levels. Doing so leads to the extreme
three level (3L) model 41 , a two parameter expression for β in this limit:
3L
3L
βxxx
→ βxxx
(ext) = f (E)G(X),
3
2
3/2
E + E+1 ,
f (E) = (1 − E)
2
"
#
r
√
3
4
G(X) =
3X
(1 − X 4 )
2
(1.9)
with
E ≡ E10 /E20 , En0 = En − E0
(1.10)
and
X≡
x01 /xmax
01 ,
xmax
01
=
~2 N
2mE10
1/2
.
(1.11)
3L
Figure 2 shows the dependence of the extreme three level βxxx
(ext) = f (E)G(X)
on the two parameters X and E. As noted above, derivation of these limits requires
truncation of the four lowest sum rules to three states, and if this truncation is
to be exact, a physical system would have to have moments and spectra satisfying
ancillary constraint conditions, viz.,
∞
X
p=3
[2Ep0 − (En0 + Em0 )] xnp xpm = 0
(1.12)
for (n, m) = (0, 0), (0, 1), (0, 2), and (1, 2), a seemingly improbable event. Also,
higher order sum rules, eg, S22 would necessarily require more than three states to
converge.
The extreme three level model is the asymptotic limit of the TLA as the three
level sum rules become exact, ie, as β → 1. In this limit, the scaling parameters
take the values (E, X) → (0, 0.79). As E → 1, the model predicts that β → 0. In
fact, if En ∼ 1/n2 , then the ratio E ∼ 27/32, suggesting that molecular structures
with Coulomb-like potentials should have poor first hyperpolarizabilities, as indeed
they all do. Though the model is only approximate when E moves away from zero,
it suggests that energies scaling inversely with an inverse power of the state number
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Fig. 2. Scaling of the extreme three level (intrinsic) first hyperpolarizability with the three level
model parameters X and E. The maximum of unity occurs when E = 0 and X = 0.79.
should have very small β, while those scaling with some positive power of the state
number should have larger β, thus corroborating the Monte Carlo results.
It has been established that maximizing the hyperpolarizabilities is of little use
in determining the potential in any model 53,54 , an important conclusion and one
that suggests a variety of potentials comprised of continuous, piecewise continuous
and even discontinuous functions might generate a large response, so long as the
spectrum is superscaling. Note that superscaling spectra need not be regular, ie,
the energies may fall between fixed boundaries that superscale with state number,
so-called root boundaries.
We also note that all nonrelativistic single and many particle Hamiltonians
studied to date have spectra and states that generate nonlinearities that are at
most 70% of the maximum for β. For the second hyperpolarizability, the apparent
limits are 60% of the fundamental limits. These appear to be hard limits. It is not
known whether systems exist that can achieve the actual fundamental limits. 19 .
But even these so-called potential limits are desirable to achieve, as they are much
larger than anything known today.
In summary, we require systems with superscaling spectra in order to have any
chance of achieving a significant fraction of the theoretical maxima of the nonlinear
optical response. We also require systems with geometries that can provide transition moments that maximize the response while simultaneously being constrained
by the TRK sum rules. We now turn to quantum graphs, which are dynamical
models of systems that exhibit the desired spectral scaling behavior with a rich
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range of tunable geometries for optimizing the transition moments as well.
1.4. Quantum graph models
With these observations, a search was begun to discover model systems with superscaling spectra that could be reasonably expected to reflect the physics of an
optimum nonlinear optical structure. We settled on quantum graphs as a near-ideal
model system. Quantum graphs are defined by starting with a quasi-one dimensional relational graph 55,56 , giving it a metric, allowing an operator to act on the
edges, and providing boundary conditions ensuring the system satisfies completeness, closure, and unitarity. A quantum graph (QG) defined this way is a general
confinement model for quasi-one dimensional electron dynamics.
Quantum graphs are idealized models of various physical and chemical systems,
devised to gain insight into problems that were otherwise analytically intractable.
They were introduced by Pauling in his study of the diamagnetic anisotropy of
aromatic compounds 57 . In this model, electrons move from one carbon atom to
another along the bonds under the influence of external fields. Generalizations were
created by Kuhn 58 , and by Ruedenberg and Scherr 59 as a free electron model
for conjugated systems. This led to a series of papers of numerical calculations 60 ,
demonstration models 61 , and a rigorous mathematical formulation of the model.
Quantum graphs have subsequently been investigated for applications to mesoscopic systems 62 , optical waveguides 63 , quantum wires 64,65 , excitations in fractals
66
, and fullerines, graphene, and carbon nanotubes 67,68,69 . Kuchment70 provides a
concise survey of the application of quantum graphs to thin structures, including
photonic crystals. In recent years, quantum graphs were shown to be exactly solvable models of quantum chaos 71,72,73,74 . Comprehensive reviews of this enormous
field are available in the literature 75,76,77 , and the mathematical rigor for these
models is quite impressive. A recent, thorough review of applications of quantum
graphs may be found in Chapter 7 of Berkolaiko and Kuchment77 .
For all of these models, the mathematics of reduction of a complex physical
problem to that of particle dynamics governed by a differential operator on the
edge of a graph is itself intractable, so heuristic arguments are often employed to
justify the use of the models. Also, models with finite transverse dimensions and
leaky modes have been studied to improve the realism of the model. As with any
such endeavor, a great deal about the original system may be learned by studying a
simplified model, including ways to improve the model and attempt to generate experimental confirmation. Microwave networks have recently been successfully used
to experimentally simulate quantum graphs 78 .
Most recently, quantum graphs have emerged as models of tightly confined
nanowire systems for nonlinear optics 79,52,80 because of their superscaling energy spectra, and their large number of topological and geometrical configurations.
Quantum graph models for nonlinear optics were initially studied to explore the
effects of extreme confinement in quantum wires 81,79 . In these models, motion in
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the dimension transverse to the wire is confined by a limiting procedure, and the
electrons are confined to dynamics along the wire. In the limit of vanishing transverse width, only the longitudinal motion contributes to the hyperpolarizabilities.
However, the TRK sum rules contain contributions from both the longitudinal and
transverse dimensions. The existence of new nonlinear optical physics on quantum
confined wires is the genesis of the quantum graph explorations, as such graphs
have a rich range of topologies and geometries with which to explore optimization
of the response.
The generalized QG model of an N electron structure constrains dynamics to
the edges of the metric graph. Dynamics are governed by a self-adjoint, multiparticle Hamiltonian operating on the edges and possessing a complete set of eigenstates
and eigenvalues. The general Hamiltonian contains momentum, position, and spin
operators, as well as functions of each describing particle-particle interactions, coupling to external fields, and other interactions. Transitions between states determine
the nonlinear optical response of the graph to an external optical field. The canonical commutation relations guarantee that the TRK sum rules hold for the transition
moments, providing constraints and relations among the various allowed transitions
in the system 41,43 .
The one-electron version of the generalized quantum graph model (hereafter
referred to as the elementary QG) is exactly solvable. Quantum graphs with zero
potential energy (bare edges) and nonzero potentials (dressed edges) have recently
been applied to calculate the off-resonance first (βijk ) and second (γijkl ) hyperpolarizability tensors (normalized to their maximum values) of elementary graphical
structures, such as wires, closed loops, and star vertices 82,79 and to investigate
the relationship between the topology and geometry of a graph and its nonlinear
optical response 52 through its hyperpolarizability tensors. The results showed that
the elementary QG model of a 3-edge star graph generated a first hyperpolarizability over half the fundamental limit and a second hyperpolarizability whose range
was between 20 − 40% of the fundamental limit. These results suggest that graphs
comprised of combinations of stars could be suitable models of moieties with the
superscaling spectra required to approach the fundamental limits, the subject of
this paper.
It is desirable to use the exactly solvable elementary QG models for exploring
the topologies yielding the largest hyperpolarizabilities, mainly because they are
simpler and tractable compared to a generalized QG model. That this may be done
is seen as follows. The generalized QG model may be expected to reflect dynamics
of a multi-electron system, e.g., a solid nanowire with a band structure and a superscaling spectrum and set of transition moments satisfying the TRK sum rules.
Since the topology of the generalized QG determines the scaling of the spectrum,
we expect the spectrum of the generalized and elementary QG models to be globally similar, though differing in details of the excited-state spectrum arising from
electron interactions. The transition moments of the generalized QG are sums of
the moments for each electron, and it should be expected that the geometry of the
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generalized QG can be manipulated to more or less match the moments of some
geometry of the single electron elementary QG. This implies that the spherical tensor components of both the generalized and elementary QG should exhibit similar
global behavior with the shape of the nanowire. Computations of the hyperpolarizability of multielectron systems show that their maxima are unchanged from those
of the single electron systems when the number of electrons is accounted for.83
Thus, the use of the elementary QG model to investigate the optimum topology for
a nonlinear optical nanostructure is a physically meaningful step.
Section 2 reviews the general methods for calculating the required spectra and
transition moments for quasi-one dimensional graphs, and the first and second
hyperpolarizability tensors in the quasi-1D limit, i.e., when the two-dimensional
calculation is reduced by taking the transverse dimension along the edges of the
graph to zero while confining the electrons with an infinite potential. The concept
of a graphical motif is introduced and discussed. Section 3 presents the hyperpolarizabilities for a variety of star-based topologies and shows how this class of graphs
can approach over 60% of the fundamental limit. It is also shown how a change
in topology can cause a dramatic change in the response by altering the states
and spectrum contributing to the hyperpolarizabilities. Section 4 summarizes the
application of the elementary QG model to elementary and composite graphs and
points to the next direction. Several Appendices contain the detailed methods for
calculating quantum graph states and spectra, show how to solve graphs when the
spectra are degenerate, and show how to scale the star motif to N ≥ 4 edges.
Two new and fundamental results emerge from this work that can aide the
molecular designer of nanowire and quantum-confined systems for nonlinear optics.
The first is that the optimum topology for quantum confined systems are those
containing star vertices, and that wires and loops are simply not capable of achieving
a large response. The second is that the star topologies generally have the largest
intrinsic responses achievable to date and might be realized in quasi-one dimensional
nanostructures having superscaling spectra and a broad range of possible transition
moments.
2. Nonlinear optics in the elementary QG model
2.1. A quasi-1D dynamical system
The dynamics of an electron on a quantum graph are described by a self-adjoint
Hamiltonian operating on the edges of the graph, with complex amplitude and probability conservation (hereafter referred to as flux conservation throughout the paper) at all internal vertices and fixed, infinite potentials at the termination vertices
(where the amplitude vanishes). The physics of the eigenstates and their spectra
have been previously described, along with a suitable lexicography for describing
the union operation for creating eigenstates from the edge functions that solve the
equations of motion for the Hamiltonian 79 .
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The graph is specified by the location of its vertices and the edges connecting
the vertices. Fig 3 details the notation and configuration of a graph. A set of vertices
with arbitrary locations in the 2D plane but fixed connections specifies a topological
class of graphs. For a fixed topology, the variation of vertex locations specifies
various geometries for the graph. Since motion is confined to the graph edges and
is continuous at each vertex, the energy spectrum depends only on the edge lengths
and the boundary conditions, ie, the topology. Spectra are quasi-quadratic in state
number, ie, superscaling. This is the first requirement for a large response from a
nonlinear optical structure.
The edge lengths and angular positions determine the projections of electron
motion onto a fixed, external reference axis. The projections summed over all edges
yield the transition moments required to compute the tensor elements of the hyperpolarizability tensors. Regardless of how the axes used to define the vertices are
chosen, the various tensor components may be used to assemble any component in
a different frame by using the rotation properties of the tensors.
The study of the nonlinear optical properties of a specific graph topology requires solving the graph for its eigenstates and spectra as functions of its edge
lengths and using them to compute a set of transition moments for the graph from
which the hyperpolarizability tensors may be computed.
Fig. 3. A four-edge quantum graph. Each edge has its own longitudinal coordinate si ranging
from zero to Li . The projection x(si ) of an edge onto the x − axis is measured from the origin
of the coordinate system attached to the graph (and not to the beginning of the edge itself). For
example, x(s1 ) = s1 cos θ1 while x(s2 ) = L1 cos θ1 + s2 cos θ2 and so on.
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2.2. States and spectra
We address so-called bare graphs in this paper. These are graphs with zero potential
energy everywhere except at their external vertices, where the potential is infinite.
A set of dressed graphs having potentials on the edges has recently been analyzed
for nonlinear optics.80 .
A general quantum graph is solved for its spectra and states once the Hamiltonian has been specified. Then, the Hamiltonian is used to compute a set of edge
functions, φin (si ) that are solutions to the Schrödinger equation with eigenvalues
En , which will be the same on all edges and identical to those for the entire graph.
Next, the edge functions are used to construct eigenstates of the Hamiltonian for
the graph through union process that reflects a direct sum Hilbert space over the
edges 79 :
i
ψn (s) = ∪E
i=1 φn (si )
(2.1)
The edge functions for an edge connecting a vertex with amplitude An to vertex
with amplitude Bn may be written in a canonical form that automatically matches
(nonzero) amplitudes at each internal vertex:
(i)
(i)
An sin kn (ai − si ) + Bn sin kn si
(2.2)
sin kn ai
Vertices with zero amplitude occur when the edges in the graph are rationallyrelated and are discussed in Section 2.4. For the rest of this paper, we assume the
edges are irrationally-related, so that the denominator in Eq. (2.2) never vanishes.
Terminal vertices with zero amplitude will take the form of Eq. (2.2) with one of
An or Bn equal to zero.
The union is defined such that the eigenstate is continuous at every vertex of the
graph, while the probability current is conserved at each vertex and thus throughout the graph. These two boundary conditions guarantee that the eigenstates are
complete. They also generate the relationships among the edge amplitudes required
to compute the spectrum of the graph.
There are always a sufficient number of equations among the coefficients such
that a characteristic equation for the eigenvalues may be extracted. To see this, let
the graph have E edges, VE external vertices, and VI internal vertices, each internal
vertex Vi having a degree di that counts the number of edges connected to vertex
Vi . The total number of vertices is V = VE + VI . The number of boundary conditions needs to be identical to the number of unknown coefficients, which is 2E (two
for each edge, according to Eq. (2.2). There are exactly VE amplitude boundary
conditions from the external vertices. There are also exactly VI flux boundary conditions from the internal vertices. So far we have exactly V boundary conditions. At
each vertex having degree di , there are exactly dki − 1 amplitude boundary condiP
tions, and summing this over all internal vertices yields i (di − 1) total amplitude
boundary conditions. The first sum is simply the number of edge endpoints connected by internal vertices, and is clearly equal to 2E − VE , while the second sum
φin (si ) =
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is the number of internal vertices, Vi . This yields exactly 2E − VE − VI = 2E − V
amplitude boundary conditions at internal vertices. The total number of boundary
conditions is obviously 2E, as required.
Solutions of the 2E coupled amplitude equations resulting from the boundary
conditions exist only if the determinant of the matrix of coefficients vanishes. This
condition produces the secular or characteristic equation for the graph and determines the eigenvalues kn and the exact energy spectrum En = ~2 kn2 /2m. Since the
boundary conditions in the elementary QG model are independent of the angles the
edges make with respect to one another, the secular equation is independent of angles and depends only on dimensionless parameters kn ai . For a given configuration
of vertices, the distance between them and the rules by which they are connected,
i.e., the topology of the graph, determines the energy spectrum. Topologically different graphs with identical geometries have different energy spectra. In this way,
the graph topology has a large impact on the nonlinear optical response.
Except for bent wires and closed loops, the secular equation of a graph is generally a transcendental equation. Accurate solutions are easily found numerically.
(i)
(i)
From these, the internal amplitudes An and Bn may be calculated relative to the
same normalization constant. Normalizing the eigenfunction produces the states
required to compute the transition moments.
It should be noted that the transition moments are sums (not unions) over edges
of the following form:
E Z ai
X
i
φ∗i
(2.3)
xnm =
n (si )φm (si ) x(si )dsi
i=1
0
where φim (si ) is the normalized wave function on the ith edge that obeys the boundary conditions for the graph and x(si ) is the x-component of si , measured from the
origin of the graph (and not of the edge), and is a function of the prior edge lengths
and angles. With edge wave functions of the form in equation (2.2), the computation of the transition moments requires integrals of products of sines and cosines
with either s or 1, all of which are calculable in closed form 79,52,82 .
We note that the transverse wavefunction is not calculated and not essential
in this model. The transverse states do not contribute to the hyperpolarizabilities
81
. But there are residual effects from the transverse state in the sum rules, as has
been previously discussed 81,79 .
The process for calculating the spectra and transition moments of elementary
QG’s can be summarized as follows: (1) select a particular graph topology, specifying the number of vertices and the connecting edges, (2) generate a random set of
vertices, and calculate the lengths of the edges and the angles each makes with the
x-axis of the graph’s coordinate system, (3) solve the Schrödinger equation on each
edge of the graph, and (4) match boundary conditions at the vertices and terminal
points. This results in a set of equations for the amplitudes of the wavefunctions
on each edge. The solvability of this set requires that the determinant of the amplitude coefficients vanishes, leading to a secular equation for the eigenvalues. The
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transition moments xnm and energies En = ~2 kn2 /2m may be used to compute the
first and second hyperpolarizabilities of any graph specified by a set of vertices, as
described next.
2.3. Hyperpolarizability tensors
For the quasi-1D problem the tensors are indexed in the (x,y) directions. The full
tensor expressions are given as a sum over states. We choose to normalize energies
and transition moments to provide a concise, dimensionless expression for both
hyperpolarizabilities. The first intrinsic hyperpolarizability tensor for 2D graphs
may then be written as
3/4 X i j k
′ ξ ξ̄
β
3
0n nm ξm0
βijk ≡
=
,
(2.4)
βmax
4
en em
n,m
i
where ξnm
and en are normalized transition moments and energies, defined by
i
ξnm
=
i
rnm
,
max
r01
en =
En0
,
E10
with r(i=1) = x and r(i=2) = y, Enm = En − Em , and where
1/2
~2
max
.
r01
=
2mE10
(2.5)
(2.6)
max
r01
represents the largest possible transition moment value of r01 41 . According
to equation (2.5), e0 = 0 and e1 = 1. βijk is scale-invariant and can be used
to compare molecules of different shapes and sizes. Similarly, the second intrinsic
hyperpolarizability is given by


j j
i
k
i ¯j
k l
1  X ′ ξ0n
ξn0
ξ0m ξm0
ξnm ξ̄ml
ξl0 X′ ξ0n
.
γijkl =
−
2e
4
en em el
e
m
n
n,m
n,m,l
(2.7)
We already know that quantum graphs exhibit superscaling spectra and that
some topologies are expected to have shapes which yield large nonlinearities. Which
geometries of a given topology yield a larger response? And which topologies show
the most promise for enabling a specific geometry to have one of the larger possible
responses? This knowledge is obtained by specifying a fixed topology, such as a star,
loop, or wire, and calculating the response for a large number of possible geometries
in order to discover the best shape. By best, the experimentalist usually means the
one with the largest value of the hyperpolarizability in a lab frame whose x-axis is
known and usually used to reference the optical field polarizations interacting with
the material.
The specification of a graph through its vertices, the calculation of its states
and spectra, and the sampling of large numbers of geometries to create ensembles
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of transition moments, energies, and hyperpolarizabilities, is a Monte Carlo computation. The results of such a calculation are a set of tensors for a topological class
of graphs whose variability is solely determined by the geometrical properties of
the graphs.
The graphs are generated by randomly picking vertices and connecting them
to generate the desired topology. The transition moments and hyperpolarizability
tensors are computed in the reference frame defined by the coordinate system used
to specify the vertices. The four nonzero tensor components for βijk and the five
nonzero tensor components for γijkl represent the tensors of the graph in the x-y
frame. If x is the laboratory x-axis, then the initial graph is likely not in the correct
orientation to yield the largest x-diagonal components desired by the experimentalist. But the graph is easily rotated through an angle into the orientation yielding
the largest diagonal components by using the rotation properties of the tensors:
βxxx (φ) = βxxx cos3 φ + 3βxxy cos2 φ sin φ
+ 3βxyy cos φ sin2 φ + βyyy sin3 φ,
(2.8)
and
γxxxx (θ) = γxxxx cos4 θ + 4γxxxy cos3 θ sin θ
+ 6γxxyy cos2 θ sin2 θ + 4γxyyy cos θ sin3 θ
+ γyyyy sin4 θ,
(2.9)
where the values of φ and θ that maximize the left-hand side of either equation
usually differ, and the tensor components on the right-hand side of either equation
are referenced to zero rotation angle, ie, the original position of the graph. By
definition, βxxx (φ) (γxxxx (θ)) is at an extreme value when the graph is rotated
through φ (θ). Once the graph is solved and the tensor components are known in
its frame, φ (θ) is easily found by maximizing equation (2.8) for βxxx (φ) (equation
(2.9) for γxxxx (θ)).
The tensor norms are the full contraction of the tensors with themselves and are
invariant under any transformation by the rotation group, and provide immediate
insight into the limiting responses of the graphs. They are given by
1/2
2
2
2
2
(2.10)
|β| = βxxx
+ 3βxxy
+ 3βxyy
+ βyyy
and
2
2
2
2
|γ| = γxxxx
+ 4γxxxy
+ 6γxxyy
+ 4γxyyy + γyyyy
1/2
.
(2.11)
These are the magnitudes of the graph’s hyperpolarizabilities and are both scale
and orientation-independent. The use of tensors to extract the nonlinear optical
response as a function of geometry and topology is often accomplished by writing
the Cartesian tensors in their irreducible spherical forms using Clebsch-Gordon
coefficients in order to identify the dipole, quadrupole, octupole, etc. components
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and has been extensively discussed in the literature
to aromatic systems 86 and quantum graphs 52 .
84,85
15
, as has their application
2.4. Solving quantum graphs with motifs.
The topological properties of a quantum graph are identical to those of the corresponding non-metric (relational) graph. Such graphs contain a number of vertices,
connected by directed or undirected edges. Edges in quantum graphs have no direction. Graphs contain vertices Vi with di edges connected to them (star vertices),
where di is the degree of vertex labeled i. Graphs also contain loops and terminated
edges. A close examination of any graph shows that it may be constructed from a
few fundamental graphs called motifs.
The spectra of connected quantum graphs are the solutions to their secular
equations, which always take the form of combinations of the secular functions
of the motifs. The motifs in Figure 4 are the 3-star and lollipop graphs, and are
sufficient to compute the states and spectra for all graphs. For brevity, we limit
graphs to those containing 3-stars, though the generalization is straightforward.
An example is presented in Appendix D.
Fig. 4. The four primary motifs for constructing any graph. The dark line at the end of an edge,
such as those at the end of the edges labeled b and c in the lower right corner graph, indicates that
the edge is terminated at an infinite potential. Unterminated edges are labeled by an amplitude
variable, indicating that the edge functions there are nonzero, as is the flux flowing into or out of
the edge with that label.
The nonlinearities of both the 3-star graph with edges terminated at infinite
potential (A = B = C = 0) and the lollipop with its stick terminated at infinite
potential (A = 0) have been calculated in the elementary QG model 52 . As isolated
models of nonlinear, quantum confined systems, these are interesting structures because both topologies have intrinsic nonlinearities over half the fundamental limits.
The method of solution of arbitrary graphs comprised of the motifs is developed in
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detail in Appendix A. With that machinery, all of the graphs discussed in the next
section are computable. We move now directly to the results of the computations.
The interested reader may refer to the Appendices to learn how to compute graphs.
3. Topological optimization of quantum graphs
3.1. Topological and geometrical classes
The character of quantum graphs comprised of star and lollipop motifs is dominated
by the properties of the motifs. Star and lollipop graphs have large intrinsic first
and second hyperpolarizabilities, implying that composites containing stars and
lollipops will have topological characteristics enabling geometric realizations with
large hyperpolarizabilities. Geometric constraints can reduce the dynamic range of
the hyperpolarizability tensors by limiting the projections of the transition moments
onto a specific external axis. Further constraints, such as a closed topology with no
external edges, can significantly alter the range of response for the graph 79 .
Wires, loops, and stars have spectra that are (more or less) evenly spaced.
Wires and loops have fixed eigenvalue spacing, whereas three-prong star graphs
have fixed spacing between so-called root separators that divide the spectrum into
cells of equal width, each containing a single eigenvalue. The variation in spectra
enabled by altering the lengths of the prongs of the star graph are due precisely to
the variability between levels permitted by the root boundaries, but the achievement of any desired eigenvalue separation, such as that achieved in the Monte Carlo
studies that generated near unity maxima, is not possible in a single star graph.
But as shown in this paper, many of the composite graphs in Figure 5 have nonlinearities larger than the three-prong star. These same graphs have nonuniform root
separators, and certain topological combinations of edge lengths enable variable
level spacing that more closely resembles that achieved in the sum-rule-constrained
Monte Carlo studies. We anticipate that a sufficiently complex graph may be devised such that the level spacing of the most significantly-contributing levels could
be near-optimum for achieving the maximum nonlinearity.
Figure 5 summarizes the results of the Monte Carlo study for all of the topologies
discussed in this paper. The Figure shows classes of graphs with the same topology
but different geometry, the N-stars, and reveals that the nonlinearities are quite
similar. Figure 5 also shows a class of lollipop-like geometries. The simple lollipop,
the bullgraph, and the lollipop bull have similar nonlinearities, as their topologies
are essentially identical. But opening a corner of the lollipop turns the graph into a
3-star but with one bent prong. The star ensures the nonlinear is not small, but the
geometry reduces the nonlinearities below those for the actual 3-star. Opening the
star turns the lollipop into a bent wire graph whose nonlinearity is now that of a
wire, not a star. Similar remarks hold for the class of barbell graphs: Those with the
star motif have about the same optical response, while opening both stars converts
the graph to a wire and dramatically lowers the nonlinearity. Finally, closing both
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stars into loops produces the greatest reduction in the response because, as with
the loop graph, the ground state is now a zero energy state with constant flux
throughout the graph.
3.2. Topological clusters
Figure 6 summarizes in pictorial form the relationship between the topology of
the graph and its first and second hyperpolarizability for a collection of loops,
wires, and stars. The range for βxxx is always −1 ≤ βxxx ≤ 1, and is symmetric in
absolute value around zero. Graphs with positive βxxx may be rotated by π to yield
the identical negative value. The top figure therefore displays only positive values.
Graphs placed along common vertical lines have the same topology but different
geometry; graphs placed on the same horizontal line have the same geometry but
different topology. A close examination reveals that loop and wire graphs without
a star vertex cluster together with lower nonlinearities, while all graphs containing
a star vertex cluster with much higher nonlinearities. Yet all of these graphs have
superscaling spectra. For example, the spectrum of loops of length L is En (loop) =
2π 2 n2 /L2 , while that of a wire of length L is En (wire) = π 2 n2 /2L2 . The difference is
that the eigenstates of the loop are doubly degenerate and have a zero energy ground
state with constant flux (i.e., n = 0, ±1, ±2 . . . for loops), while the eigenstates of
wires are nondegenerate with a positive energy ground state. The loop topology
actually restricts the range of the transition moments, limiting the nonlinearities.
The wire topology has no such restriction. But here, topology limits the response
of the wire by creating cancellations in the state overlap of the transition moments,
since eigenstates must oscillate continuously across the wire, creating both negative
and positive overlap regions.
3.3. Topological shifts in similar geometries
The global properties of the hyperpolarizability tensors are thus determined by the
topology of the graphs, while the local properties, such as the projections onto a
fixed external axis, are determined by the geometry of a particular realization of the
graph. For a given Monte Carlo run on a specific topology, a complete sampling of
possible geometries yields the ranges of the first and second hyperpolarizability, as
well as the contributions to these tensors from their spherical components. Graphs
with identical shapes but different topologies necessarily have different spectra,
though the projection of their edges onto a fixed external axis could be similar.
Topological shifts alter the spectra, changing both the values and the energy-level
spacing; these factors set the limits on the maximum achievable hyperpolarizability
in the graph, even when the geometry is optimized for that graph.
To study a particular typological class, we sample its configuration space using Monte Carlo methods. The coordinates of the edges that define the shape are
selected at random; and, the intrinsic first and second hyperpolarizabilities calcu-
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Fig. 5. Intrinsic nonlinearities of topological classes of quantum graphs. The first (βxxx ) and
second (γxxxx ) hyperpolarizabilities shown are the largest values for the geometries within the
specific topological class. The first hyperpolarizability tensor norm βnorm is defined and calculated
in the text, and is an invariant for the topological class. In all cases except closed loops, the
maximum value of βxxx is equal to βnorm , indicating that the topology allows the graph to
assume its best configuration for the xxx component, which usually means that the yyy component
vanishes. Loops are so tightly constrained that it is impossible for a loop to have one of its diagonal
components at zero when the other is nonzero.
Graph
Geometry Topology
βnorm
|βxxx |
γxxxx
bent
wire
line
0.172
0.172
-0.126
to 0.007
triangle
loop
0.086
0.049
-0.138
to 0
3-star
3-fork
0.58
0.58
-0.138
to 0.3
4-star
4-fork
0.53
0.53
-0.125
to 0.27
5-star
5-fork
0.51
0.51
-0.11 to
0.26
6-star
6-fork
0.51
0.51
-0.11 to
0.26
7-star
7-fork
0.51
0.51
-0.11 to
0.26
lollipop
starloop
0.62
0.62
-0.12 to
0.20
bull
starloop
0.53
0.53
-0.09 to
0.20
lollipop
bull
starloop
0.51
0.51
-0.09 to
0.19
lollipop
3-fork
0.33
0.33
-0.11 to
0.13
lollipop
line
0.17
0.17
-0.09 to
0.006
barbell
2-fork
lollipop
0.54
0.54
-0.104
to 0.214
barbell
dual
2-fork
0.43
0.43
-0.13 to
0.22
barbell
starloop
0.41
0.41
-0.07 to
0.11
barbell
line
0.14
0.14
-0.085
to 0.006
barbell
loop
0.11
0.11
-0.1 to
0.002
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Geometrical changes
Topological changes
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
xxx
Geometrical changes
Topological changes
-0.2
-0.1
0.0
0.1
0.2
0.3
xxxx
Fig. 6. Complete range of the first (top) and second (bottom) hyperpolarizability (horizonal axis)
for the loop, wire, and star topologies (vertical axis) with various geometries for each. The vertical
bins that change with βxxx (and γxxx ) show that graphs with similar topologies have essentially
the same hyperpolarizability.
lated. The distribution of results over many configurations provides insights into
the relationship between a topological class and its nonlinear-optical properties.
Figure 7 illustrates the approach when applied to two distinct topologies that
span the same geometries: Two realizations of a barbell, one containing closed bells
and the other having two open bells. The former contains two 3-star vertices but
the stars are closed into loops, and the entire graph is sealed, as explained earlier in
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the paper. The open barbell graph has two open stars connected in such a way that
flux travels across the structure, rather than around the loops. The energy levels of
the graphs are quite different, and so are the hyperpolarizability tensors.
0.6
Dual 2-fork
Barbell
| |
0.4
0.2
0.0
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.05
0.10
0.15
xxx
0.15
Dual 2-fork
Barbell
| |
0.10
0.05
0.00
-0.15
-0.10
-0.05
0.00
xxxx
Fig. 7. Hyperpolarizability tensors and their norms for the barbell graph with two open ends
(star-to-star) and two closed ends (bells) for a large sampling of shapes using a Monte Carlo
method. The profound change in the nonlinear response due to the topological change from a
closed dual-loop configuration to a geometrically similar one that is isomorphic to two back-toback 3-star graphs is self-evident.
Figure 8 applies the same approach to two distinct topologies that span the
same lollipop geometry. The lollipop graph contains a star vertex and a loop, and
it has the largest first hyperpolarizability predicted for any of the star-containing
graphs in this paper. The other a topological four-wire graph. The energy levels of
the graphs are quite different, and so are the hyperpolarizability tensors.
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0.8
Lollipop
Open lollipop
| |
0.6
0.4
0.2
0.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
xxx
0.3
Lollipop
Open lollipop
| |
0.2
0.1
0.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
xxxx
Fig. 8. Hyperpolarizability tensors and their norms for the lollipop graph and 4-wire lollipop-like
graph for a large sampling of shapes using a Monte Carlo method. The profound change in the
nonlinear response due to the topological change from a star-based configuration to a geometrically
similar one that is a bent wire graph is self-evident.
3.4. Geometrical tuning
Graphs containing star motifs have superscaling spectra, but their wavenumbers are
no longer uniformly spaced. Instead, the wavenumbers fall between evenly spaced
root boundaries 87 . Energies have a quasi-quadratic spectrum, with level spacings
that may be tuned by altering the geometrical properties of the graph. Figure 9
illustrates the impact of edge length variations in 3-star graphs. The relative edge
lengths set the energy spectrum of the graph and also contribute to their projection
onto an external x-axis once their angular positions are specified. For a given set
of prong lengths, the value of βxxx will vary over a range as the angles between
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the prongs change. However, for each set of prong lengths, there will be one set of
angles for which βxxx is maximum. The figure was constructed so that the largest
values were plotted on the top. For example, stars with prongs (1, 0.6, 0.13) appear
to have the largest βxxx , but this is true only if the angles take on specific values.
Underneath the contours showing the greatest values for this set of prong lengths,
there are points with lesser values corresponding to the same prong lengths but
suboptimum angles; this is evident from Figure 10. The inset in Figure 9 shows the
shapes with the largest values (red), as well as one with a much smaller value(blue).
1.0
Smallest prong's length
0.000
0.8
0.1000
0.2000
0.3000
0.6
0.4000
0.5000
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Middle prong's length
Fig. 9. Contour plot of the largest values of βxxx for 3-star graphs as a function of prong lengths.
The largest prong always has length unity (since the results are scale-independent). The length of
the middle prong ranges from zero to unity, while that of the shortest prong ranges from zero to a
maximum equal to the middle prong. The angles each makes with the longest prong are random.
Each pair (short, middle) of prong lengths has a set of angles where βxxx is near zero, but only
the optimum pairs (short, middle) can generate large βxxx for special sets of angles. The inset
shows the shape with the largest (red) and smallest (blue) βxxx .
The ideal star graph has its longest prong (of length one) and its second longest
prong (of length ∼ 0.6 antiparallel along x and the shortest prong (of length ∼ 0.13)
at any angle. Interestingly, a straight wire along x would have zero βxxx , while a bent
wire could have βxxx ∼ 0.172 but no larger. The attachment of a single short prong
away from a center of symmetry converts the graph to a topology that generates
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0.6
0.1
0.15
0.5
0.2
0.3
0.4
xxx
0.4
0.5
0.3
0.2
0.1
0.0
0.00
0.25
0.50
0.75
1.00
Angle (units of 2 ) between mid and long prongs
Fig. 10. Variation of βxxx with the angle between the middle prong of length 0.6 and the long
prong of length unity for several short prong lengths. The vertical range of points for a specific
curve representing a short prong length is for the full range of angles of the small prong. The key
feature determining the strength of the nonlinearity is the antiparallel middle and large prongs,
with a short prong at any angle. A short prong permits the largest flux to move across the graph
without diverting any of it into another direction. Increasing the short prong length dramatically
decreases the nonlinear response. At a short prong length of 0.3 or greater, the nature of the
angular dependence changes.
one of the largest intrinsic values to date.
The star topology ensures that the flux may be distributed such that state overlap for the transition moments is maximized while flux is simultaneously conserved.
The third prong in the graph mimics a discontinuity across the other two prongs
due to a finite δ potential. The addition of a prong to a wire to create a star is topologically equivalent to dressing a wire with a finite δ potential, a so-called dressed
graph that has recent been analyzed 80 .
We conclude that the effect on the hyperpolarizabilities of topological changes
in quantum graphs is to induce spectral changes that favor higher response as well
as allow geometrical tuning that can optimize the response through shape. This
latter point was carefully studied in Ref 52 for a class of loop graphs and a 3-star
using a spherical tensor method 84,85,88 .
3.5. Approaching the fundamental limits
Superscaling spectra derived from a potential energy yield first hyperpolarizabilities
with a maximum of 0.7089 rather than unity 46,89,53,80 . If only three states were
required for the sum rule, the extreme three level model would be exact, and we
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would have E = 0.5 but with the same (universal) value of X ∼ 0.79. However,
these models always require more than three states to satisfy the sum rules, as they
must when the maximum is below unity. Figure 11 is a dressed quantum graph
which nearly hits the maximum value of βxxx = 0.7089, while Fig. 12 illustrates
the phenomena 80 . For this example, E converges to a value of about 0.45. The sum
rules require a minimum of four states to converge, though the three level model is
nearly exact at the maximum. This is a general feature of nonlinear optical systems
whose origin has been recently analyzed and discussed in detail 90 .
Fig. 11. A quasi-one dimensional quantum graph with a variable potential (g/L)δ(s) located
between the endpoints. This dressed graph has one of the largest nonlinearities (0.705) of any
structure to date, nearly equal to the potential limit of 0.7089.
4. Conclusions
The ideal energy spectrum for maximizing the nonlinear optical response of any
quantum structure scales as some positive power of the state number. This has been
previously established by Monte Carlo simulations valid for any non-relativistic
many body Hamiltonian with random spectra and transition moments constrained
by sum rules. Heuristically, we argued that the theory of fundamental limits is
also highly suggestive that so-called superscaling spectra are required if the optical
nonlinearities are to bridge the large (factor of 30) gap between all presently known
intrinsic molecular responses and the fundamental limit.
Quantum graphs–dynamical models of electrons confined to the edges of a mathematical graph on which a Hamiltonian operates–have exactly the superscaling
spectra required to maximize the nonlinear optical response. The distribution of
energies is set by the graph topology, while the geometry may be altered to change
the transition moments in an advantageous way. Topology also affects the transition
moments, in that it determines the flow of electrons in the graph and the shape of
the eigenstates. State shape and state overlap determine the signs and magnitudes
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1.0
|
|
0.8
|
3L
|
f(E)G(X)
S
S
S
0.6
S
S
00
00
00
00
00
(3)
(4)
(5)
(6)
(7)
0.4
0.2
g=-3.73
a=0.28
0.0
-10
-5
0
5
10
g
Fig. 12. Scaling of the full, three level, and extreme three level expressions for βxxx of a linear
system with a δ potential of strength g. The three level model becomes exact at the maximum
of βxxx , which is the three level Ansatz in action. The extreme three level model is nearly 20%
larger at the maximum, however. Also shown are the values of the truncated sum rule S00 with
3,4,5,6, and 7 terms. The deviation of the models is clearly due to the requirement that the sum
rule requires more than three terms to converge.
of the transition moments. Consequently, graphs having distinct topological classes
are expected to have similarly sized nonlinear optical responses.
In this paper, we presented a model of the generalized one electron quantum
graph as a nonlinear optical structure, showed how to compute the hyperpolarizabilities, and invoked a general motif method to aid in calculating the spectra and
states of more complex graphs containing star vertices, as well as loops and wires.
We provided a set of rules for calculating any graph, taking into account both degeneracies from rationally-related edges as well as the appearance of multiple sets of
eigenstates arising from subgraphs. We also related the global properties of closed
graphs to the appearance of a zero energy, constant amplitude ground state. The
quantitative results of a large Monte Carlo study of these graphs were presented in
Section 3, with the conclusion that graphs containing star motifs have the largest
optical nonlinearities of any model structure known to date.
The general methods for solving for the hyperpolarizabilities of quantum graphs
that we previously developed could then be used with the states and energies from
the motif analysis to calculate the cartesian and spherical tensor components of the
first (β) and second (γ) hyperpolarizability to understand the impact of topology
across geometrically equivalent graphs on the nonlinear optical tensors. In particular, graphs with identical topologies have comparable maximum nonlinearities,
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while graphs with identical geometries but different topologies have far different
maximum nonlinearities. This behavior has been previously observed for bent wires
and loops 79 so it is not surprising that it holds for star graphs and their extensions. But with the advent of the star motif for constructing the spectral equations
for complex graphs, we now have a fundamental explanation for both the similar, topological responses and the differences when topologies are altered so that
the underlying secular spectral functions of geometrically similar graphs no longer
resemble one another. Scaling according to the theory of fundamental limits also
holds across different star geometries, so long as the star motif is active within the
graph so that its global properties are dominated by the star topology. Interestingly,
the addition of a star vertex to a loop creates the lollipop graph which has one of
the largest intrinsic first hyperpolarizabilities of all graphs, despite the fact that
the loop by itself has a nonlinearity that is over ten times smaller. The star vertex
is key to the synthesis of molecular systems modeled by the elementary quantum
graph, as it appears to guarantee that a geometrically-unconstrained star topology
will have a large, intrinsic first and second hyperpolarizability.
The one-electron elemental graph model is a simple but effective way to explore
a wide range of states and transition moments enabled by a structure’s Hamiltonian
and boundary conditions, from the bottom up, i.e., by solving the equations of motion to determine the maximum hyperpolarizabilities of a topological class of graphs
for comparison with the abstract theory of fundamental limits based upon the use
of the Thomas-Reiche-Kuhn sum rules in a sum over states expansion of β and γ.
We have argued that multi-electron models will differ vastly in their construction
and the richness of their physics, but that their spectra and transition moments
should reflect the global properties of the one-electron model. Consequently, we
expect the results presented in this paper to be valid for actual quantum confined
systems.
Acknowledgments
SS and MGK thank the National Science Foundation (ECCS-1128076) for generously supporting this work.
Appendix A. Solution of quantum graphs using motifs
The results in the paper were computed using a new method for solving an arbitrary
quantum graph for its states and spectra.
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A.1. Star graph motifs
The conservation of flux in a star graph leads to the reduced secular function fstar
, where
52,87
fstar (ai ) =
E
X
cot kn ai
(A.1)
i=1
for an E-pronged star with edges ai .
For the 3-star with edges a, b, c, multiply the reduced secular function by
sin kn a sin kn b sin kn c, a factor that is nonzero for irrationally-related edges, and
we get the secular function Fstar (a, b, c):
1
[cos kn L1 + cos kn L2
4
+ cos kn L3 − 3 cos kn L] ,
Fstar (a, b, c) =
(A.2)
where L = a + b + c, L1 = |a + b − c|, L2 = |a − b + c|, and L3 = |a − b − c|.
We call equation (A.2) the canonical form of the 3-star secular function and will
use it extensively in what follows. The combination lengths are equivalent to the
edge lengths, and we freely move back and forth between them. For example, a
star graph with edges d, e, f will have a secular function Fstar (d, e, f ) which may
be written in the form of the right hand side of equation (A.2) with the set (a, b, c)
replaced by (d, e, f ) in the definition of the combination lengths.
The nature of the solutions to the secular equation Fstar = 0 for irrational
lengths have been discussed at length in Ref. 87 , where a periodic orbit expansion
was derived for the eigenvalues. They are nondegenerate and lie one to a cell between
root boundaries at multiples of π/L. For our purposes, a set of solutions for any
finite number of wave functions is easily found by numerically intersecting the two
parts of the secular equation. In this way, a set of nondegenerate eigenvalues may
be obtained for arbitrary (but irrational) prong lengths. Solutions may be found in
Ref. 52 . The degenerate case is solved later in this section.
In Figure A1, the ten lowest eigenvalues are displayed for a Monte Carlo run
where the ensemble members are ordered such that their maximum βxxx increases
from left to right. The numerical value of βxxx is displayed as the dashed curve
and its value shown on the right axis in Figure A1. The eigenvalues each vary in
a random way between their root separators as the geometry of the star is altered
from left to right until the optimum geometries are attained. When the maximum
is approached, the lowest eigenvalues converge to well-defined values. The ratio
E ≡ E10 /E20 takes the value 0.4 for star graphs at their maximum. Recall from
Section 1 that this three level model parameter is a measure of how close the graph
is to an optimum.
The final piece required to calculate hyperpolarizabilities are the transition moments. These require the eigenstates for the graph, which are calculated from the
individual edge states, per Eq. (2.3). The edge states for the star graph have been
detailed in Ref. 52 .
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12
1.0
11
0.9
10
0.8
9
0.7
8
n
k L/
6
0.5
5
xxx
0.6
7
0.4
4
0.3
3
0.2
2
0.1
1
0
0.0
0
2000
4000
6000
8000
10000
Run #
Fig. A1. Variation of the lowest ten eigenvalues of the 3-star graph for 10, 000 randomly generated
samples, ordered by their βxxx values. For the 3-star, the solutions to the secular equation lie
between fixed root boundaries located at multiples of π/L, with L equal to the sum of all edges. The
three lowest eigenvectors asymptote to fixed values as βxxx (shown as a dashed curve) approaches
its maximum value for the best geometry.
Consider now how to use the motifs to construct the secular functions for composite graphs containing stars. For the star graph with three terminated ends, the
secular function Fstar is exactly zero. When the ends in the motifs are unterminated, the amplitudes at the ends are nonzero and there must be flux moving in or
out of these ends, since flux is conserved in the graph. This means that the secular
function is no longer zero but is related to the flux entering or leaving the unterminated vertices. For the fully unterminated 3-star motif in Figure 4, the canonical
form of the edge functions is
Zn sin kn (a − sa ) + An sin kn sa
sin kn a
Zn sin kn (b − sb ) + Bn sin kn sb
φn (sb ) =
sin kn b
Zn sin kn (c − sc ) + Cn sin kn sc
φn (sc ) =
sin kn c
φn (sa ) =
(A.3)
where each distance s on an edge is measured from the central vertex. For unterminated ends, conservation of flux at the central vertex Z produces the following
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secular equation relating the amplitudes at the ends and the central amplitude:
Zn Fstar (a, b, c) = An sin kn b sin kn c
+ Bn sin kn a sin kn c
(A.4)
+ Cn sin kn a sin kn b
The right-hand side is the net flux through its unterminated vertices required
to conserve flux at the central vertex. If the ends are terminated, the righthand side vanishes, reproducing the secular equation for a terminated star graph,
Fstar (a, b, c) = 0. For unterminated ends, equation (A.4) relates the amplitudes at
the ends and at the central vertex through a single equation.
We may use the 3-star motif to compute the secular equation of a graph consisting of two 3-star motifs. Consider the graph in Figure A2 with two star vertices
connected by a common prong. There are two central vertices connected by an edge,
and each is a 3-star motif with two ends at zero amplitude. The coupled amplitude
equations are easy to write down using equation (A.4) for each star and appropriate
relabeling the vertex amplitudes and edges to match those of the composite graph.
They are
An Fstar (a, b, e) = Bn sin kn a sin kn b
Bn Fstar (c, d, e) = An sin kn c sin kn d
(A.5)
The secular function for this graph is thus
Fstar−star = Fstar (a, b, e)Fstar (c, d, e)
− sin kn a sin kn b sin kn c sin kn d
Fig. A2.
The hybrid star-to-star graph formed from the union of two star motifs.
(A.6)
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The secular function in equation (A.6) may be rewritten in the following form:
Fsec = −4 sin kn a sin kn b sin kn c sin kn d sin kn e
− 2 sin kn (a + b + c + d + e) + sin kn (a + b − c − d + e)
− sin kn (a + b − c − d − e) + .5 sin kn (a + b + c − d + e)
+ .5 sin kn (a + b + c − d − e) + .5 sin kn (a + b − c + d + e)
(A.7)
+ .5 sin kn (a + b − c + d − e) + .5 sin kn (a − b + c + d + e)
+ .5 sin kn (a − b + c + d − e) − .5 sin kn (a − b − c − d + e)
− .5 sin kn (a − b − c − d − e)
Using motifs, we have obtained the secular function for this back-to-back star
graph in a few steps. The amplitudes at the two vertices are easily calculated from
Eq. (A.5). The transition moments and hyperpolarizabilities are computed using
the machinery from Section 2. The generalization to graphs comprised of many
stars is straightforward.
A.2. Lollipop motifs
For the lollipop graph, the secular function Fpop (a, Ltot ) is 52
Ltot
1
3 cos kn a +
Fpop (a, Ltot ) =
2
2
Ltot
.
− cos kn a −
2
(A.8)
where Ltot = b + c + d is the length of the loop and a is the prong length.
The wavefunctions of the lollipop graph are a composite of two sets of wavefunctions, one set that is nonzero at the central vertex and on all edges, and one for
wavefunctions that vanish at the origin and are exactly zero on the prong edge. The
first set correspond to the symmetric wavefunctions of a 3-sided bent wire (open at
the central vertex) coupled to a nonzero prong wavefunction, while the second set
correspond to the asymmetric wavefunctions of a 3-sided bent wire (open at the
central vertex) with a zero prong wavefunction. When another graph is attached
to the prong, the loop-only wave functions go away and we’re left with the wave
functions satisfying the secular equation above 52 .
The spectrum of the star motif had uniform root boundaries between which
all eigenvalues were found for any geometry of the graph. This observation is not
true for the lollipop graph, whose spectrum is a complex interleaving of two sets
of disparate spectra as discussed above. The spectrum for a Monte Carlo run of
lollipops with variable geometry is illustrated in Figure A3, ordered by increasing
βxxx . There are always well-defined boundaries between a given set of eigenstates
for a fixed run, and that somewhere between runs 3000 and 4000, βxxx begins to
climb, and the states jump to a different-looking pattern where the variation of the
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three lowest eigenvalues decreases rapidly and then converge to fixed values at the
maximum hyperpolarizability, with a universal value of the energy ratio E ∼ 0.4.
Since the maximum value of βxxx for lollipops is larger than that of the basic
star graph, we might expect that further changes in the complexity of the spectrum
of a graph could lead to even larger responses. In complex graphs, the root boundaries may acquire an almost random structure to them, suggesting they might be
tunable to provide the kinds of level spacing required to achieve maximum nonlinear
responses.
1.0
7
0.9
6
0.8
n
0.7
0.6
4
0.5
3
xxx
k (a+L/2)/
5
0.4
0.3
2
0.2
1
0.1
0.0
0
0
2000
4000
6000
8000
10000
Run #
Fig. A3. Variation of the lowest eight momentum eigenvalues of the lollipop graph for 10, 000
randomly generated samples, ordered by their βxxx values. βxxx is shown as the dashed curve.
Consider now the unterminated lollipop in Figure 4. The exact expression for
the flux in/out of the lollipop motif is
Zn Fpop (a, Ltot ) = An cos kn Ltot /2.
(A.9)
The left-hand side is the total flux exiting the central vertex Z and entering the
vertex A. When A = 0, the flux conservation equation becomes Fpop (a, Ltot ) = 0.
This determines the eigenvalues of the terminated lollipop graph where there is
flux moving on all of its edges but never exiting at vertex A. As noted above, the
terminated lollipop has an additional spectrum comprised of wave functions where
there is exactly zero flux on edge a at all times, i.e., flux just circulates around the
loop. This set must be included in the total spectrum of the lollipop.
We are now in a position to use the secular functions for the star and lollipop
motifs to solve for the secular equation of the combined graph in Figure A4. We
see from the Figure that we should connect the lollipop to the star such that vertex
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Z of the star in Fig 4 is vertex A of the lollipop in the same figure. Relabeling the
vertices and edges to match those of the composite graph, we get
An Fstar (a, b, c) = Bn sin kn b sin kn c
(A.10)
Bn Fpop (a, Ltot ) = An cos kn Ltot /2
Cross-multiplying (or setting the determinant of the coefficients to zero) yields the
secular function Fpop−star (a, b, c, d, e) for the eigenvalues of the star-stick lollipop
graph:
Fpop−star (a, b, c, d, e) = Fstar (a, b, c)Fpop (a, Ltot )
(A.11)
− sin kn b sin kn c cos kn Ltot /2
The solutions to Fpop−star (a, b, c, d, e) = 0 are the eigenvalues of the star-stick
lollipop graph. The amplitudes An and Bn are then found from Eq. (A.10). With
these in hand, the hyperpolarizabilities for this class of graphs are calculated as
described previously in Section 2.
Fig. A4.
The hybrid star-lollipop graph formed from the union of the star and lollipop motifs.
Appendix B. Computational rules for general graphs
Every graph may be solved using the fundamental star and lollipop motifs in the
same way as done for the previous composite graphs. We wish to note some general
rules for using motifs to solve graphs. We will again limit the discussion to internal
vertices of degree equal to three or less, but the generalization to internal vertices of
arbitrary degree is straightforward and requires use of the N-star motifs for degree
N.
The general method to writing down a secular equation for a graph with VI internal vertices is as follows: (1) Label the vertices with their amplitudes A, B, C, . . .
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where flux flowing into or out of each is conserved and must flow along edges connected to the vertex and to other parts of the graph, (2) determine how the vertex
and its edges overlays other vertices and their connected edges in order to identify
the motifs comprising the graph, (3) use the secular functions in equation (A.4)
and equation (A.9) with appropriately relabeled amplitudes to write a set of simultaneous equations relating the secular functions at an internal vertex to the
connecting amplitudes via the motif equations, and (4) set the determinant of the
amplitude matrix to zero to obtain the secular equation for the entire graph. The
secular equation of a graph is generally transcendental but is easily solved using
numerical methods.
The secular equation provides a set of eigenvalues for the states of the graph
where the amplitude of the particle on each edge is nonzero. Graphs containing
closed loops, such as the lollipop or barbell, can also have wavefunctions where the
amplitude along a connecting or terminal edge is exactly zero, as was described for
the lollipop motif in this section. When multiple sets of wavefunctions are present,
they must be ordered in energy and their eigenstates interleaved so that a complete
set results for the graph. Finally, graphs with no external connections, such as a
barbell or triangle, will necessarily have a zero-energy eigenstate where the wavefunction over the entire graph is constant. This ground state must be included in the
spectrum in order to maintain completeness of the eigenstates. For most composite
graphs, there will not be any additional sets of spectra other than those from the
secular equation. Again, the rational case is an exception, allowing wavefunctions
that vanish at the shared vertices and form exact half-periods over each edge. These
are straightforward to handle, should they arise, and do not require solution to any
transcendental secular equation.
We realize that the rules presented here are not nearly as expeditious as are
Feynman rules for particle interactions. But they are general and enable any graph
to be analyzed rapidly. It will be true, however, that as the size of the graph grows,
it will be challenging to solve the resultant secular equation, as there are likely
to be many configurations with degenerate states. Degeneracies require a bit of
care to deal with, and are discussed next. Moreover, it will be necessary to track
the metric distance of each edge from a common origin in order to compute the
transition moments. For graphs with a dozen edges, this isn’t too challenging, but
one can imagine a graph having twenty edges in a rather irregular configuration
being quite difficult to solve. Fortunately, we are seeking quantum graphs with large
nonlinearities that also reflect the physics or chemistry of simple molecular systems,
and we have already discovered that the basic star and lollipop graphs have large
nonlinearities.
Appendix C. Handling degeneracies
Throughout this paper, the edges of the stars have been constrained to be
irrationally-related. This ensures the canonical form of the edge functions in equa-
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tion (2.2) may be used without reservation, as the edge functions never vanish
at the central vertices. We briefly examine rationally-related edges to show how
to solve these, too. The isolated star graph can have doubly-degenerate states for
wavenumbers satisfying kn = nπ/L, with L = a + b + c for certain values of the
edges. If we write the edge functions for the three prongs as
φ(1)
n (s1 ) = An sin kn (a − s1 )
φ(2)
n (s2 ) = Bn sin kn (b − s2 )
(C.1)
φ(3)
n (s3 ) = Cn sin kn (c − s3 )
then the amplitudes at the center satisfy
An sin kn a = Bn sin kn b = Cn sin kn c,
(C.2)
and conservation of flux yields the secular function for an isolated star graph as
Fstar = An cos kn a + Bn cos kn b + Cn cos kn c.
(C.3)
If none of the sine functions in equation (C.2) vanishes, then the wavenumbers
satisfy the usual secular equation, (A.2). The derivative of the secular function is
− dFstar /dk = aAn sin kn a + bBn sin kn b + cCn sin kn c.
(C.4)
For irrationally-related edges, Fstar (kn ) = 0 determines the nondegenerate eigenvalues kn , and the derivative dFstar /dk is never zero for k = kn . But when the edges
are rationally-related, both the secular equation and its derivative will occasionally
vanish for the same k, the doubly-degenerate eigenvalues. When this occurs, Eq.
(C.3) and Eq. (C.4) may be used to extract amplitudes for a pair of orthogonal,
degenerate states corresponding to the same eigenvalue, because the same secular equation holds for the degenerate case, as well 87 , as can be shown through a
scattering matrix solution or simply by noting that the transition from irrationallyrelated edges to rationally-related ones is equivalent to an infinitesimal change in
the arguments of Eq. (C.3). Consequently, one may move from one case to the other
by performing all the divisions used to derive Eq. (A.2) and then taking the rational limit. To see how this comes about, examine the spectrum in Figure D1. The
vertical lines at kn = nπ/L (with L = a + b + c) are the root separators, defining
cells in which only one root may be found. For certain values of the edges, there are
roots on either side of a root separator that converge toward each other and meet
at a separator (becoming degenerate roots) as the edge values are tweaked toward
specific ratios. When this happens, all three terms in Eq. (C.2) vanish, and they
also vanish in the derivative of the secular equation (which is why the roots are
doubly-degenerate). A single degenerate root has a pair of eigenstates whose amplitudes are determined by the secular Eq. (C.3) and the requirement that the pair of
degenerate states are orthogonal. If the edge coefficients are labeled (A1 B1 C1 ) and
(A2 B2 C2 ), the orthogonality condition is aA1 A2 + bB1 B2 + cC1 C2 = 0. A suitable
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set of coefficients may then be determined from this and the secular relations, with
A1 = 1, A2 = 1, C1 = 1 as the roots converge to kr as follows:
cos kr c + cos kr a
cos kr b
a cos kr b − bB1 cos kr a
C2 = −
c cos kr b − bB1 cos kr c
cos kr a + C2 cos kr c
B2 = −
cos kr b
B1 = −
(C.5)
where the cosines will take the values ±1 as their arguments each approach their
own multiple of π. Even when there are no degeneracies for a given set of rationallyrelated edges, it is possible that one or two of the sine functions in equation (C.2)
could vanish. (If all three vanish, then the root is a degenerate root boundary).
In this case, the amplitudes may still be obtained by using the amplitude equation
(C.2) and the secular equation. For example, suppose a given solution to the secular
equation km satisfies sin km a = 0, but that sin km b 6= 0 and sin km c 6= 0. Then (C.2)
and the secular equation yield the singlet solution set
sin km c
sin km b
sin km (b + c)
= −Bm
cos km a sin km c
Bm = Cm
Am
(C.6)
(C.7)
The single unknown coefficient Cm is determined by normalization, of course. When
two sine functions vanish, say sin km a = 0 and sin km b = 0, the solutions are even
easier to obtain. Then (C.2) yields Cm = 0 and Am = −Bm cos km b/ cos km a. This
solves the degenerate case for any relationship among the edges. The extension of
this analysis to graphs comprised of star motifs is straightforward, but it provides
no additional information to that obtained from the nondegenerate case.
Appendix D. Scaling to N ≥ 4 star vertices
Referring to Figure D2, we easily generalize equation (A.3) to arrive at the 4-edge
equivalent of equation (A.4) and get
Zn F4star (a, b, c, d) = An sin kn b sin kn c sin kn d
+ Bn sin kn a sin kn c sin kn d
(D.1)
+ Cn sin kn a sin kn b sin kn d
+ Dn sin kn a sin kn b sin kn c
where the secular function for the 4-star graph is given by the four-edge version of
equation (A.1), rationalized by the denominators. After some simple algebra, the
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1.5
1.0
0.5
0.0
-0.5
-1.0
F
-1.5
1.5
star
dF
star
/dk
1.0
0.5
0.0
-0.5
-1.0
-1.5
0
2
4
6
8
10
k L/
n
Fig. D1. Solutions to the secular equation of a 3-star graph with prong lengths that are
irrationally-related (upper panel) but approach a rational relationship (lower panel). The two
roots enclosed by the dotted circles in the irrational case (one each on either side of a root separator) coalesce into a pair of degenerate roots (on a root separator) enclosed by the dotted circle, as
the lengths become rationally-related The transition is smooth in that there is no abrupt change
in the nonlinear response of the graph as the edges become rationally-related. Units on the y-axis
are arbitrary.
secular function may be written as
1
[sin kn (a + b) cos kn (c − d)
2
+ cos kn (a − b) sin kn (c + d)
F4star (a, b, c, d) =
− sin kn (a + b + c + d] .
(D.2)
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This form may be used as a motif to solve composite graphs with several fouredge vertices, such as the bubble graph shown in Figure D2, for which the secular
function may again be written down by inspection:
F4bubble (a, b, c, d, e1 , e2 , f1 , f2 ) =
(D.3)
F4star (a, b, L1 , L2 )F4star (c, d, L1 , L2 )
− sin kn a sin kn b sin kn c sin kn d(sin kn L1 + sin kn L2 )2
where L1 = e1 +e2 , L2 = f1 +f2 are the two (sequential) bubble edges. Generalizing
Fig. D2. Left: 4-star motif with four open edges carrying flux to/from another motif to which
their edges might be attached. Right: a bubble graph comprised of two 4-star motifs.
to any number of edges is straightforward. We note here that adding prongs to a
3-star simply changes its geometry, not its topology. The global behavior of the
nonlinearities is therefore similar to that of the 3-star.
References
1. R. H. Baughman and R. R. Chance. Fully Conjugated Polymer Crystalls: Solid-State
Synthesis and Properties of the Polydiacetylenes. Annals of the New York Academy
of Sciences, 313:705–725, 1978.
2. G. F. Lipscomb, A. F. Garito, and R. S. Narang. An exceptionally large linear electrooptic effect in the organic-solid MNA. J. Chem Phys., 75(3):1509–1516, 1981.
3. P. G. Huggard, W. Blau, and D. Schweitzer. Large Third-Order Optical Nonlinearity of the Organic Metal x-[Bis(Ethylenedithio) Tetrathiofulvalene] Triiodide. Appl.
Phys.Lett., 51(26):2183–5, 1987.
4. D. S. Chemla. Non-Linear Optical Properties of Condensed Matter. Rep. Prog. Phys.,
43:1191–1262, 1980.
5. J. A. Giordmaine. Nonlinear Optical Properties of Liquids. Phys. Rev., 138(6A):A
1599– A 1606, 1965.
6. J. A. Giordmaine. Nonlinear Optical Properties of Liquids. J. Chem. Phys.,
64(215):A1599–A1606, 1967.
7. P. P. Ho and R. R. Alfano. Optical Kerr Effect in Liquids. Phys. Rev. A, 20(5):2170–
87, 1979.
8. C. H. Chen and M. P. McCann. Measurements of Two-photon Absorption Cross
Sections for Liquid Benzene and Methyl Benzene. J. Chem. Phys., 88(8):4671–7, 1988.
9. M. I. Barnik, L. M. Blinov, A. M. Dorozhkin, and N. M. Shtykov. Optical Second
Harmonic Generation in Various Liquid Crystalline Phases. Mol. Cryst. Liq. Cryst.,
98:1–12, 1983.
10. D. P. Shelton and A. D. Bucking. Optical Second-Harmonic Generation in Gases with
a Low-Power Laser. Phys. Rev. A, 26(2):2787–2798, 1982.
May 5, 2014 0:15 WSPC/INSTRUCTION FILE
38
ws-jnopm
Rick Lytel, Shoresh Shafei, Mark Kuzyk
11. P. Kaatz, E. A. Donley, and D. P. Shelton. A comparison of molecular hyperpolarizabilities from gas and liquid phase measurements. J. Chem. Phys., 108(3):849–856,
1998.
12. U. Gubler, Ch. Bosshard, P. Gunter, M. Y. Balakina, J. Cornil, J. L. Bredas,
R. E. Martin, and F. Diederich. Scaling law for second-orderhyperpolarizability in
poly(triacetylene) molecular wires. Opt. Lett., 24:1599, 1999.
13. M.A. Foster, A.C. Turner, M. Lipson, and A.L. Gaeta. Nonlinear optics in photonic
nanowires. Opt. Exp.,, 16(2):1300–1320, 2008.
14. B. Tian, P. Xie, T.J. Kempa, D.C. Bell, and C.M. Lieber. Single-crystalline kinked
semiconductor nanowire superstructures. Nat. Nanotech., 4(12):824–829, 2009.
15. D.S. Chemla and D.A.B. Miller. Room-temperature excitonic nonlinear-optical effects
in semiconductor quantum-well structures. J. Opt. Soc. Am. B, 2(7):1155–1173, 1985.
16. DAB Miller, DS Chemla, TC Damen, AC Gossard, W. Wiegmann, TH Wood, and
CA Burrus. Band-edge electroabsorption in quantum well structures: The quantumconfined Stark effect. Physical Review Letters, 53(22):2173–2176, 1984.
17. S. Schmitt-Rink, D. S. Chemla, and D. A. B. Miller. Linear and Nonlinear Optical
Properties of Semiconductor Quantum Wellls. Adv. Phys., 38(2):89–188, 1989.
18. S. Schmitt-Rink, DAB Miller, and DS Chemla. Theory of the linear and nonlinear
optical properties of semiconductor microcrystallites. Phys. Rev. B, 35(15):8113, 1987.
19. Mark G. Kuzyk, Javier Perez-Moreno, and Shoresh Shafei. Sum rules and scaling in
nonlinear optics. Phys. Rep, 529:297–398, 2013.
20. P. D. Maker and R. W. Terhune. Study of Optical Effects Due to an Induced Polarization Third-Order in the Electric Field Strength. Phys. Rev., 137(3A):801–818,
1965.
21. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee. Optical SecondHarmonic Generation in Reflection from Media with Inversion Symmetry. Phys. Rev.,
174(3):813–22, 1968.
22. M. Bass, D. Bua, and R. Mozzi. Optical second-harmonic generation in crystals of
organic dyes. App, 15:393–396, 1969.
23. R.W. Waynant and M.N. Ediger. Electro-optics handbook. McGraw-Hill, 2000.
24. L.W. Tutt and T.F. Boggess. A review of optical limiting mechanisms and devices
using organics, fullerenes, semiconductors and other materials. Progress in quantum
electronics, 17(4):299–338, 1993.
25. A. Yariv and D.M. Pepper. Amplified reflection, phase conjugation, and oscillation in
degenerate four-wave mixing. Optics letters, 1(1):16–18, 1977.
26. A. Yariv. Phase conjugate optics and real-time holography. IEEE J. Quant. Elec.,
14(9):650–660, 1978.
27. Robert W. Boyd. Nonlinear Optics. Academic Press, 1992.
28. R. Lytel. Pump-depletion effects in noncollinear degenerate four-wave mixing in kerr
media. J. Opt. Soc. Am. B, 3:1580–1584, 1986.
29. H.G. Winful and J.H. Marburger. Hysteresis and optical bistability in degenerate
four-wave mixing. Applied Physics Letters, 36:613, 1980.
30. H.M. Gibbs. Physics of optical bistability (a). J. Opt. Soc. Am. A, vol. 1, page 1281,
1:1281, 1984.
31. A. Weiner. Ultrafast optics, volume 72. Wiley, 2011.
32. R. Lytel. Optical multistability in collinear degenerate four-wave mixing. J. Opt. Soc.
Am. B, 1:91–94, 1984.
33. TE Van Eck, AJ Ticknor, RS Lytel, and GF Lipscomb. Complementary optical tap
fabricated in an electro-optic polymer waveguide. Appl Phys Lett, 58(15):1588–1590,
1991.
May 5, 2014 0:15 WSPC/INSTRUCTION FILE
ws-jnopm
Optimum topology of . . .
39
34. J. Zyss. Nonlinear Organic Materials for Intergrated Optics: A Review. J. Mol. Electron., 1:25–45, 1985.
35. R. W. Boyd. Nonlinear Optics. Academic Press, 3rd edition, 2009.
36. M. G. Kuzyk. A bird’s-eye view of nonlinear-optical processes: Unification through
scale invariance. Nonl. Opt. Quant. Opt., 40:1–13, 2010.
37. L. Hornak. Polymers for lightwave and integrated optics: technology and applications.
Marcel Dekker, Inc, 270 Madison Ave, New York, New York 10016, USA, 1992. 768,
1992.
38. M. G. Kuzyk. Polymer Fiber Optics: materials, physics, and applications, volume 117
of Optical science and engineering. CRC Press, Boca Raton, 2006.
39. B. J. Orr and J. F. Ward. Perturbation Theory of the Non-Linear Optical Polarization
of an Isolated System. Molec. Phys., 20(3):513–526, 1971.
40. S. Wang. Generalization of the thomas-reiche-kuhn and the bethe sum rules. Phys.
Rev. A, 60(1):262–266, 1999.
41. M. G. Kuzyk. Physical Limits on Electronic Nonlinear Molecular Susceptibilities.
Phys. Rev. Lett., 85:1218, 2000.
42. M. G. Kuzyk. Using fundamental principles to understand and optimize nonlinearoptical materials. J. Mat. Chem., 19:7444–7465, 2009.
43. M. G. Kuzyk. Fundamental limits of all nonlinear-optical phenomena that are representable by a second-order susceptibility. J. Chem Phys., 125:154108, 2006.
44. M. G. Kuzyk. Fundamental limits on third-order molecular susceptibilities: erratum.
Opt. Lett., 28:135, 2003.
45. J. Zhou, M. G. Kuzyk, and D. S. Watkins. Pushing the hyperpolarizability to the
limit. Opt. Lett., 31:2891, 2006.
46. J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk. Optimizing potential
energy functions for maximal intrinsic hyperpolarizability. Phys. Rev. A, 76:053831,
2007.
47. J. Pérez-Moreno, Y. Zhao, K. Clays, and M. G. Kuzyk. Modulated conjugation as a
means for attaining a record high intrinsic hyperpolarizability. Opt. Lett., 32(1):59–61,
2007.
48. J. Pérez-Moreno, Y. Zhao, K. Clays, M. G. Kuzyk, Y. Shen, L.. Qiu, J. Hao, and
K. Guo. Modulated conjugation as a means of improving the intrinsic hyperpolarizability. J. Am. Chem. Soc., 131:5084–5093, 2009.
49. M. Belloni and R. W. Robinett. Quantum mechanical sum rules for two model systems. Am. J. Phys., 76(9):798–806, 2008.
50. M. G. Kuzyk. Truncated Sum Rules and their use in Calculating Fundamental Limits
of Nonlinear Susceptibilities. J. Nonl. Opt. Phys. & Mat., 15(1):77–87, 2006.
51. S. Shafei and M. G. Kuzyk. Critical role of the energy spectrum in determining the
nonlinear-optical response of a quantum system. J. Opt. Soc. Am. B, 28:882–891,
2011.
52. Rick Lytel, Shoresh Shafei, Julian H. Smith, and Mark G. Kuzyk. Influence of geometry and topology of quantum graphs on their nonlinear optical properties. Phys. Rev.
A, 87:043824, Apr 2013.
53. TJ Atherton, J. Lesnefsky, GA Wiggers, and RG Petschek. Maximizing the hyperpolarizability poorly determines the potential. J. Opt. Soc. Am. B, 29:513–520, 2012.
54. C. J. Burke, T. J. Atherton, J. Lesnefsky, and R. G. Petschek. Optimizing the second hyperpolarizability with minimally parametrized potentials. J. Opt. Soc. Am. B,
30(6):1438–1445, Jun 2013.
55. Béla Bollobás. Modern graph theory, volume 184. Springer, 1998.
56. Mark EJ Newman. The structure and function of complex networks. SIAM review,
May 5, 2014 0:15 WSPC/INSTRUCTION FILE
40
ws-jnopm
Rick Lytel, Shoresh Shafei, Mark Kuzyk
45(2):167–256, 2003.
57. Linus Pauling. The diamagnetic anisotropy of aromatic molecules. J. Chem. Phys.,
4:673, 1936.
58. H. Kuhn. Free Electron Model for Absorption Spectra of Organic Dyes. J. Chem.
Phys., 16:840–841, 1948.
59. Klaus Ruedenberg and Charles W Scherr. Free-electron network model for conjugated
systems. i. theory. The Journal of Chemical Physics, 21(9):1565–1581, 1953.
60. Charles W Scherr. Free-electron network model for conjugated systems. ii. numerical
calculations. The Journal of Chemical Physics, 21(9):1582–1596, 1953.
61. John R Platt. Free-electron network model for conjugated systems. iii. a demonstration model showing bond order and“free valence”in conjugated hydrocarbons. The
Journal of Chemical Physics, 21(9):1597–1600, 1953.
62. D Kowal, U Sivan, O Entin-Wohlman, and Y Imry. Transmission through multiplyconnected wire systems. Physical Review B, 42(14):9009, 1990.
63. C Flesia, R Johnston, and H Kunz. Strong localization of classical waves: a numerical
study. EPL (Europhysics Letters), 3(4):497, 1987.
64. EL Ivchenko and AA Kiselev. Electron g factor in quantum wires and quantum dots.
Journal of Experimental and Theoretical Physics Letters, 67(1):43–47, 1998.
65. JA Sánchez-Gil, V Freilikher, I Yurkevich, and AA Maradudin. Coexistence of ballistic
transport, diffusion, and localization in surface disordered waveguides. Physical review
letters, 80(5):948, 1998.
66. Y Avishai and JM Luck. Quantum percolation and ballistic conductance on a lattice
of wires. Physical Review B, 45(3):1074, 1992.
67. Claudio Amovilli, Frederik E Leys, and Norman H March. Electronic energy spectrum
of two-dimensional solids and a chain of c atoms from a quantum network model.
Journal of mathematical chemistry, 36(2):93–112, 2004.
68. Frederik E Leys, Claudio Amovilli, and Norman H March. Topology, connectivity,
and electronic structure of c and b cages and the corresponding nanotubes. Journal
of chemical information and computer sciences, 44(1):122–135, 2004.
69. Peter Kuchment and Olaf Post. On the spectra of carbon nano-structures. Communications in Mathematical Physics, 275(3):805–826, 2007.
70. Peter Kuchment. Graph models for waves in thin structures. Waves in Random Media,
12(4):R1–R24, 2002.
71. Tsampikos Kottos and Uzy Smilansky. Quantum chaos on graphs. Phys. Rev. Lett.,
79:4794–4797, Dec 1997.
72. T. Kottos and U. Smilansky. Periodic orbit theory and spectral statistics for quantum
graphs. Ann. Phys., 274(1):76–124, 1999.
73. Tsampikos Kottos and Uzy Smilansky. Chaotic scattering on graphs. Physical review
letters, 85(5):968, 2000.
74. R. Blümel, Y. Dabaghian, and RV Jensen. One-dimensional quantum chaos: explicitly
solvable cases. JETP Lett, 74:258, 2001.
75. Pavel Exner. Analysis on Graphs and Its Applications: Isaac Newton Institute for
Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007, volume 77. American Mathematical Soc., 2008.
76. Gregory Berkolaiko. Quantum Graphs and Their Applications: Proceedings of an
AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their
Applications, June 19-23, 2005, Snowbird, Utah, volume 415. American Mathematical
Soc., 2006.
77. Gregory Berkolaiko and Peter Kuchment. Introduction to quantum graphs. Number
186. American Mathematical Soc., 2013.
May 5, 2014 0:15 WSPC/INSTRUCTION FILE
ws-jnopm
Optimum topology of . . .
41
78. Oleh Hul, Szymon Bauch, Prot Pakonski, Nazar Savytskyy, Karol Zyczkowski, and
Leszek Sirko. Experimental simulation of quantum graphs by microwave networks.
Physical Review E, 69(5):056205, 2004.
79. S. Shafei, R. Lytel, and M. G. Kuzyk. Geometry-controlled nonlinear optical response
of quantum graphs. J. Opt. Soc. Am. B, 29:3419–3428, 2012.
80. Rick Lytel and Mark G Kuzyk. Dressed quantum graphs with optical nonlinearities
approaching the fundamental limit. Journal of Nonlinear Optical Physics & Materials,
22(04), 2013.
81. S. Shafei and M. G. Kuzyk. The effect of extreme confinement on the nonlinear-optical
response of quantum wires. J. Nonl. Opt. Phys. Mat., 20:427–441, 2011.
82. Rick Lytel, Shoresh Shafei, and Mark G Kuzyk. Nonlinear optics of quantum graphs.
In Proc. SPIE 8474, pages 84740O–84740O–9, 2012.
83. D. S. Watkins and M. G. Kuzyk. The effect of electron interactions on the universal properties of systems with optimized off-resonant intrinsic hyperpolarizability. J.
Chem. Phys., 134:094109, 2011.
84. J. Jerphagnon, D. Chemla, and R. Bonneville. The description of the physical properties of condensed matter using irreducible tensors. Adv. Phys., 27(4):609–650, 1978.
85. T. Bancewicz and Z. Ozgo. Irreducible spherical representation of some fourth-rank
tensors. J. Comput. Meth. Sci. Eng., 10(3):129–138, 2010.
86. M. Joffre, D. Yaron, J. Silbey, and J. Zyss. Second Order Optical Nonlinearity in
Octupolar Aromatic Systems. J . Chem. Phys., 97(8):5607–5615, 1992.
87. ZS Pastore and R. Blümel. An exact periodic-orbit formula for the energy levels of
the three-pronged star graph. J. Phys. A, 42:135102, 2009.
88. J. Zyss and I. Ledoux. Nonlinear optics in multipolar media: theory and experiments.
Chem. Rev., 94(1):77–105, 1994.
89. Shoresh Shafei and Mark G. Kuzyk. Paradox of the many-state catastrophe of fundamental limits and the three-state conjecture. Phys. Rev. A, 88:023863, Aug 2013.
90. Mark G Kuzyk. A heuristic approach for treating pathologies of truncated sum rules
in limit theory of nonlinear susceptibilities. arXiv preprint arXiv:1402.3827, 2014.
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